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Dry spreading of polymer solutions
M. Boudoussier
To cite this version:
M. Boudoussier. Dry spreading of polymer solutions. Journal de Physique, 1987, 48 (3), pp.445-455.
�10.1051/jphys:01987004803044500�. �jpa-00210460�
Dry spreading of polymer solutions
M. Boudoussier
Collège de France, Physique de la Matière Condensée (*), 11 Place Marcelin-Berthelot, 75231 Paris Cedex 05, France
(Requ le 16 mai 1986, accept6 le 8 octobre 1986)
Résumé. - Nous discutons l’étalement d’une solution semi-diluée de chaînes neutre et flexible sur une surface solide, en supposant que : a) le solvant est bon et n’est pas volatile (mouillage sec) ; b) le polymère ne s’adsorbe sur
aucune interface (solide/liquide et liquide/air). Le polymère dissout a deux principaux effets : a) il modifie le
coefficient d’étalement du liquide, b) il introduit une contribution nouvelle à la pression de disjonction, qui, comme
l’a remarqué de Gennes [7] n’est présente que si le polymère ne peut pas s’échanger avec un réservoir. Nous montrons ici que cette pression de disjonction stabilise le film pour le mode d’ondulation q = 0, mais ne contribue pas à la stabilité du film face aux ondulations à q ~ 0 : chaque moitié de longueur d’onde d’une ondulation joue un rôle
de réservoir pour l’autre moitié. Cette stabilité marginale suggère une transition de phase. Nous analysons l’énergie
libre d’une gouttelette complètement étalée, prenant en compte les forces de Van der Waals et les contributions du
polymère, et nous construisons un diagramme de phase comprenant trois états différents : a) la gouttelette non étalée
de solution, b) le film de solution contenant le polymère, c) le film de solvant pur sans polymère. La gouttelette non
étalée peut exister seulement au-dessus d’une certaine concentration en polymère øw. Quand les forces de
Van der Waals sont faibles, la gouttelette non étalée peut être entourée d’un film de solvant pur.
Abstract. - We discuss the spreading of a semi-dilute solution of neutral, flexible chains on a solid surface, assuming that a) the solvent is good and is non-volatile (dry spreading) and b) the polymer does not adsorb on either
interface (solid/liquid and liquid/air). The polymer solute has two main effects : a) it modifies the spreading
coefficient of the liquid and b) it introduces a new contribution to the disjoining pressure, which, as pointed out by
de Gennes [7], is present only when the polymer cannot exchange with a reservoir. We show here that this disjoining
pressure stabilizes the film for the q = 0 mode of undulation, but does not contribute to the stability of the film
against undulations at q ~ 0 : each half wavelength of one undulation plays the role of a reservoir for the other half.
This marginal stability is suggestive of a phase transition. We analyse the free energy of a completely spread droplet,
incorporating both Van der Waals forces and polymer contributions, and construct a phase diagram involving three
different states (a) bulk droplet of solution, (b) solution film containing polymer and (c) film with no polymer. The droplet cannot spread only above a certain polymer concentration øw. When the Van der Waals forces are weak, the
balloon should be surrounded by a film of pure solvent.
Classification
Physics Abstracts
64.75 - 68.15 - 81.15L
1. Introduction.
1.1 GENERAL AIMS. - The wetting properties of a liquid are often drastically modified by additives : surfactants or polymers [1]. In the present paper, we focus our attention on the simplest type of polymer
additive : neutral, flexible chains in good solvent con-
ditions. We are not concerned with surfactant effects,
and we assume that our polymer does not adsorb either
on the solvent/solid boundary, or on the free surface.
This restriction is strong, but is clearly needed in a first
theoretical study: the opposite case with adsorption brings in deep complications.
(*) Unitd Associde au C.N.R.S. (U.A. 792).
What happens if we put a droplet of solution, whith a certain polymer volume fraction 0, on a solid surface ? We assume that the solvent is not volatile (dry spreading), and that it spreads spontaneously with a positive spreading coefficient [1-3]
where ysv, YSLI 1’LY represent the solid/air, sol- id/solvent, and solvent/air interfacial tensions.
The solution has a different set of tensions [4] :
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004803044500
and a new spreading coefficient
It is important to realize that AYSL and AYLV are positive, because we assumed no adsorption of the polymer on any interface : the interfacial energies are
raised by polymer contacts. Thus S is smaller than So.
We shall be concerned mainly with situations of small
So (which lead to relatively thick wetting films). Then
the correction is very important : in fact, at a certain concentration Ow, we expect S to change sign. At higher concentrations, the solution will not spread, and
it will display a finite contact angle (J.
Solutions with concentrations below ow may spread
in the form of a film. But they may also phase separate, because the polymer chains dislike to be confined in a narrow region. Our main aim is to discuss these phase separations.
First, we must construct the free energy for the various partners involved a) films of pure solvent, b) bulk solution and solution films. In section 2, we
describe the various partners (bulk solution or films) in
terms of a mean-field theory. We give numerical results
for the free energy at arbitrary thickness and closed form results in two simple limits (thin or thick). We also
show (in appendix B) that the solution films have a very
special type of stability : polymer confinement stabilizes the film against a uniform deformation, but it does not prevent other fluctuation modes ! This observation did suggest to us the possible onset of certain phase
transitions.
In section 3, we analyse the phase diagram, as
obtained from mean-field free energies. The major
control parameters are (a) the mean-volume fraction of the polymer in the original droplet, (b) dimensionless ratio built with the Hamaker constant (measuring the strength of VW energies), the spreading coefficient of pure solvent and the thermal energy kT. It should be
pointed out that our treatment of VW forces is very crude : we assume that the long-range interactions between solid and solvent and between solid and
polymer are identical. This omits certain segregation effects, which could be quite interesting.
In section 4, we try to go beyond mean-field theory,
and we set-up a scaling picture [6] for the polymer
contribution to the free energy. We can only use it in
two limiting cases [7] (thick or thin films). Then, we
consider phase separations in this scaling approach.
The results are similar to those obtained in a mean-field
approach, except that now the control parameters are
slightly simpler (they reduce to the mean-volume fraction and the ratio of Hamaker constant to the kT).
But, there is a lot of uncertainty in the scaling results
because we analyse only the limits of thick or thin films
(in fact the most stable phasis may be of intermediate
thickness).
1.2 SPREADING OF A PURE LIQUID. - The situation is shown in figure 1. If the spreading coefficient
So is positive, the liquid spreads in the form of a film of thickness e, area X, because the capillary forces tend to
make the film thinner. The competing long-range
forces are taken into account through a correction term
P (e) depending on the thickness (the disjoining
pressure is 7T = - dP/de). We restrict our attention to non-polar liquids with long-range interactions of the non-retarded Van der Waals (VW) form [8] :
A is the Hamaker constant, it has the dimension of an
energy and is often comparable to kT in magnitude.
Incorporating capillarity and long-range forces effects the free-energy of the film is neglecting edge effects :
Fig. 1 - - Film of pure liquid on a solid surface. The volume 12 of the liquid is well-defined. The edge effects concern a healing zone of size A negligible in front of the film size. Thus the surface of the film is X = 121e.
The free-energy due to the liquid/air and liquid/solid
interfaces is (vLS+ ’YLV) I because we neglect the
edge effects. The free-energy due to the solid/air
interface is ’Y sv ( Is - I) where Xs is the total area of
the solid surface. Let us call Fo the sum of the terms independent of I and e : the bulk free-energies and Is ysv. Since the volume f2 = Ye is fixed, minimiza-
tion with respect to e leads to [3] :
The resulting thickness ev,,, = (A /4 ff S,,) " is relative-
ly large if the spreading coefficient So is small. Omitting Fo the free-energy of the film is F_ =
We focus our attention on cases of small So where e - 100 A allows for a meaningful
continuum description.
1.3 SEMI-DILUTE FILMS: ENERGIES. - The liquid is
now a semi-dilute solution in good solvent with deple-
tion on both interfaces (solid-solution and air-solution).
As shown in reference [9], if polymer molecules can be exchanged with a reservoir, they do not contribute to the stability of the film. Under compression, the solute polymer moves out of the film, going into the reservoir.
Here, we consider a film where the polymer is trapped.
We may still derive the thickness e of the pancake from equation (1.3) incorporating the polymer contribution in this equation.
1.4 MARGINAL STABILITY OF SEMI-DILUTE FILMS. -
First, let us consider a small fluctuation 8e ( x ) (Fig. 2)
on a pure liquid pancake. Its stability implies the
balance between capillar pressure and disjoining
pressure :
A characteristic length appears :
v
We call it the healing length [3] since it is the size of the
healing zone on the edge of the pancake (Fig. 1).
Consider now a polymer solution film with a modu- lation 8 e = 5 eo cos qx ( q 0 0 ). Here, each half wave-
Fig. 2. - A fluctuation on the interface air-liquid of a film.
The equilibrium thickness of the film is e. The thickness fluctuation is called 8e ( x ). It can be due to a defect or the neighbourhood of an edge (healing zone).
length acts as a reservoir for the other ones. Thus, the polymer contribution to the right-hand side of (1.4) is
zero : the presence of polymer stabilizes the q = 0 mode, but not the other modes ! We conclude that (a) long-range forces, even very small, play an essential
role in the film stability, and (b) that new forms of films may appear (Fig. 8) : a film of pure solvent coexisting
with a film or a non-spread droplet containing the polymer.
2. Films of semi-dilute solutions : a mean-field ap-
proach.
We consider a film of a solution containing chains of
monomers of size b. We assume that the chains are very
entangled (semi-dilute solution), in a very good solvent (the excluded-volume is b3). We shall calculate the thickness of this film as a function of the mean-volume fraction 0 of monomers, and we shall determine its
stability. To achieve this program, we must determine the free-energy of the film. It takes into account three effects : capillarity, long-range forces, repulsion interac-
tions between monomers.
Let us first evaluate the last contribution, in the framework of a mean-field theory, before studying the
state of the film.
2.1 CONTRIBUTION OF POLYMER TO THE FREE- ENERGY [9]. - We assume that the polymer is repelled by both interfaces solution-solid and solution-air. Thus the volume fraction profile cp (z) must verify the boundary conditions :
where e is the thickness of the film (Fig. 3). We
construct a mean-field theory, following the principles
set by Edwards [5] and introducing a field variable
t/J (z) = J fJ (z) . The free-energy per unit area due to the presence of a polymer is :
Fig. 3. - Semi-dilute solution film with repulsion at both
interfaces. Near the interfaces, the concentration goes to zero
(in a mean-film approach, its first derivative too). At the
middle of the film the volume fraction has a maximum
01.
The first term takes into account the inhomogeneities
of the solution, the second term takes into account the excluded-volume effects (proportional to the square of
the concentration). The profile qi ( z ) is determined by
minimization of :¡ pol’ with a constraint of fixed number of monomers. Here, operating at fixed e, this is equivalent to a condition of fixed mean-volume frac- tion :
This leads to introduce a Lagrange multiplier A which is
the difference between the chemical potential of the
monomers and that of the solvent molecules. For
simplicity, we call it the exchange potential. Thus we
are led to minimize :
,k is determined by equation (2.3).
The minimization of j gives us the first integral:
where 0 1 = i# ( e/2 ) is the extremum value of 0 in the
centre of the film. Let us introduce dimensionless variables, by dividing the lengths by the correlation
length 1 = b I J6 .p1:
-
Taking into account (2.5), we may replace the inte- gration in (2.4) with respect to z by an integration with
respect to qi. Expressed with the new variables, we obtain :
Since we have two unknowns (qi 1 and x), we must write
down a second equation, expressing the fact that the thickness e is imposed :
The integrals are elliptic functions (to see this, we set
y = cos 0 to transform to the usual shape).
Equations (2.8) and (2.7) give x and qi 1 -1/0= respect- ively and we deduce :F pol from:
We calculate the various integrals numerically and in
the same way we solve the various equations. We
obtain the curves in figure 4 for Ypol and t/J l’ However,
we can obtain simple analytic results in two limit cases :
thick films ( e > C ) and thin films ( e ) . We define § as the correlation length that the solution would have if it were homogeneous (no wall effects) :
9 Thick films : e > (0 ) :
If the thickness e is large enough, the volume fraction
is constant in the central zone of the film q, l’ The
concentration profile in the depletion zones is indepen-
dent of e, but shows a correlation length , = b/ v/6- 0-, (see Fig. 5). So we may think that the
exchange potential is Tq,l. Let us expand (2.8) around
x == 1, we find
By e large enough, we mean e > 6 so that we can neglect the exponential corrections as in (2.11). Then
the fixed number of monomers per unit area is pro-
portional to
Relation (2.5) shows that the coefficient w is equal to
B/2. We may also write the energy as a bulk term,
proportional to e, plus a surface correction, pro-
portional to i :
Fig. 4. - Results of mean-field calculations : (a) Ratio of the volume fraction in the middle of the film 0, 1 to the mean-
volume fraction 46 as a function of elf ( 0 ) . It slowly
decreases from 2 to 1, corresponding to a bulk solution. (b) :F pol is the contribution to the surface free-energy due to the
- - ’2
polymer. ’We represent as a function of
elf ( 0 ) . Note the asyniptote of slope 1/2 corresponding to
the bulk solution.
Fig. 5. - Two simple limit cases: 8 Thick films : near the interfaces, the polymer is partly expelled from a zone of size
c ( 4, ) - The volume fraction profile in these zones is independent of the thickness e, and between them there is a
plateau.
9 Thin films : in a mean-field approximation, the repulsions
between monomers become negligible in front of the repulsion
of the interfaces (which here is supposed to be infinite). The
volume fraction profile is a period of a sinusoid.
Two methods can be used to calculate the coefficient
a:
a) use Gibbs’ relation to relate the free-energy to the
surface depletion (displayed in Eq. (2.11)),
b) equate two expressions of the chemical potential,
one locally calculated in the centre of the film and another one globally calculated
-
The complete calculation of a is in appendix A. The
result is a = 4/3 with mean-field theory.
. Thin films : e ..-c ( 0 ) :
Following equation (2.8), this corresponds to the
limit x -+ oo . Expanding this equation, we find
By (2.7), we deduce and finally :
2.2 STATE OF THE FILM. - Let us call n the volume of the film and X its area ( 12 = Ie) . To obtain the total free energy of the film, we add the contribution of the
polymer discussed above and the contributions of
i) capillarity (it takes into account the interactions between the liquid and the surfaces on a molecular
scale. Since we have depletion, this term is - So I
where So is the spreading coefficient of pure solvent)
and ii) long-range forces. We consider only non-re-
tarded Van der Waals forces. We assume that the wall-
monomer and the wall-solvent molecule interactions
are identical : the VW term is the same as in pure solvent XA/12 lTez. This is clearly an oversimplification
but it may be a reasonable starting point since we
assume a good solvent : the polarisabilities of the
solvent and of the polymer must be comparable.
The total free energy of the film is
Neglecting edge effects, we write I = 12 le, 12 is a
constant (dry spreading) (non volatile solvent, no exchange with a reservoir). The result of the previous paragraph may be expressed in the form
where f is a dimensionless function. In this problem, we
have four parameters (So, A, T, b ). Considering the
form of F, we can eliminate two parameters, introduc- ing two control parameters :
The thickness eyw = -".,,/A/4 irso corresponds to a film
of pure solvent (Sect. I). Following section 2.1 we
have calculated the function f numerically. Thus at
fixed volume fraction T, we adjust the thickness e, so as to minimize F. Considering equation (2.17), we con-
struct the curves e/evw as a function of ;Tlsm and
FL3 2/3
fi Ts4l3 as a function of 7/S , for e such that F is a
nTs / ’
minimum (Fig. 6). Then, we remark that these curves
depend only on als’13, following the expression of F in equation (2.17).
2.3 SOME COMMENTS ABOUT THESE RESULTS. -
o At T = 0, we have a film of pure solvent [3]. Thus
we find that the thickness is e_ = b J als (see Sect. 1)
and the free energy is
9 At a certain volume fraction, which we call Ow, we find that the thickness becomes infinite. Since the volume of the polymer solution is constant, the approximation of negligible edge effects stops to be
Fig. 6. - Final results of mean-field calculations for various values of the Hamaker constant A and the spreading coeffi-
cient So. The relevant parameter is (A/4 1TT)
valid near 0,,. In fact an infinite thickness means that the polymer solution does not spread on the solid and
that we have a non spreading droplet of solution (see Fig. 7). Let us show that the spreading coefficient S becomes negative above cfJw.
The spreading coefficient is modified by the presence of polymer, because of the depletion zones. Near cfJ = cfJ w’ the film is thick so we may apply the
discussion of this case of section 2.1. Then let us expand
Fig. 7. - Unspread droplet of polymer solution, in case of depletion. Because of the depletion, the spreading coefficient is lowered, it is small enough, so it can be negative. Then the polymer solution droplet forms a « balloon » (we neglect gravity, i.e. we suppose the size smaller than 1 mm (see
Ref. [3]), with a contact angle 0, determined by Laplace law [1, 3].
w The thickness e : the solution stops to spread on the solid at
- ( S, b 2/ T ) -3.
. The free-energy F : for F = 0,,, the curves join. Demix-
tions may happen (dashed lines).
9 The first term is the bulk free-energy of the polymer solution.
. The second one describes a modification of the
spreading coefficient S due to the polymer. We can
write S as :
. The other terms modify the disjoining pressure.
If T -- 0,,, S is positive and the existence of the film
is possible. The equilibrium thickness e increases indefi-
nitely when T approaches 0,,. If ;T ::. 0,,, S is negative,
and there is no film, but a non-spreading droplet of
solution. Its total energy is1 T 12 T 2. The contact angle 2
(see Fig. 7) is determined by S :
Y LV depends slightly on T. Let us remark that at 0 = 0,,, F and its first derivative are continuous. This
point will be important when we shall study the stability
of films.
